COURSE TITLE: MATRIX ALGEBRA FOR ECONOMICS COURSE CODE: ECON 2015 (EC 24B) LEVEL: UNDERGRADUATE LEVEL (SECOND YEAR) NO OF CREDITS: 3 PREREQUISITES: ECON 1001, ECON 1002, ECON 1003, COURSE DESCRIPTION / RATIONALE: This course will provide economics students with the tools required to undertake mathematical analysis in their field. The course covers a wide range of topics including mathematical induction and linear programming. The course will be taught in 2 hour lectures and then 1 hour tutorial where worksheets will be looked over and any additional questions will be answered. This is an analytical course that is all quantitative. The course will be assessed through two means as there will be a Mid-term Exam which accounts for 40% of students’ grades and then the final exam which will make up the next 60% of the grade. This course fits well with the mission of the Department and the University in terms of its contribution to preparing students with an in-depth knowledge of Matrices Algebra. PURPOSE OF THE COURSE The course is designed to; establish elementary skills in Mathematical Methods and to initiate the development of an ability to apply these mathematical methods to problems in the field of economics. INSTRUCTOR INFORMATION LECTURER: Dr Roger Hosein and Mr. Carlos Hazel EMAIL CONTACT: [email protected]

[email protected]

PHONE CONTACT: 662-2002 EXT 83041 and 662-2002 EXT 83068 Office hours- To be announced on first day of lectures Communication policy – Please contact lecturer during assigned office hours LETTER TO THE STUDENT Dear Students, Matrix Algebra is a course that will allow you to get a better understanding of practical issues of modeling in economics. This course is going to be very interactive and it will be to your benefit to participate during lectures as your questions is what will guide the sessions and make the class more interesting. The lecturer will apply the material and techniques taught to issues relating to economic principles and theories. Note that your success with the material depends on you. Welcome to Matrix Algebra for Economics. I look forward to your participation and engagement.

2 CONTENT 1. Operation on Vectors & Matrices 2. Determinants of Matrices 3. Equivalence 4. The Inverse of a Matrix 5. Vectors 6. Solving Linear Equations 7. Eigenvalues and Eigenvectors 8. Symmetric and Skew-Symmetric Matrices 9. Linear Programming: Graphical method 10. Linear programming: Simplex Method GOALS/AIMS To equip students with an adequate set of tools; theoretical and practical; to understand the application of economic principles.

GENERAL OBJECTIVES To simplify basic mathematical tools into practical easy to follow steps. LEARNING OUTCOMES Students will be able, 1. Solve different equations using Matrix Algebra 2. Perform two different methods of Linear Programming. 3. Use Matrix Algebra to solve economic problems.

COURSE ASSESSMENT 40% Mid Term Examination 60% Final Examination TEACHING STRATEGIES The mode of teaching will be via face to face lectures and tutorials sessions. RESOURCES Lecture notes will be provided to the class in the form of a compact disk. These can be collected at the Office of Dr Roger Hosein. READINGS Lecture notes prepared by Mr. Martin Franklin and Dr. Roger Hosein

3

Topics

COURSE CALENDAR Reference

Operation on Vectors & Matrices Introduction Matrix Addition Matrix Multiplication Scalar and Vector Multiplication Multiplication by a Scalar Distributive Laws and Associative Laws of Multiplication of Matrices Equality of Matrices Transpose of a Matrix Symmetric Matrices The Zero Matrix Identity and Diagonal Matrices Upper Triangular (UTM), Lower Triangular (LTM) and Diagonal Matrices Orthogonal Matrices Some Properties of Orthogonal Matrices Invertible Matrices Power of Matrices Differences between Scalars and Matrices Determinants of Matrices Evaluating the Determinant Matrix of Minors Matrix of Cofactors Laplace Expansion Theorem Properties of Determinants Other Worked Examples Equivalence Rank of a Matrix Elementary Transformations and their Inverses Elementary Transformations (ET) Inverse Elementary Transformation (IT) Equivalent Matrices Echelon Forn and Reduced Echelon From The Normal Form of a Matrix Echelon Matrices and the Rank of a Matrix The Inverse of a Matrix The Adjoint Matrix (Aadj) Inverse of a Matrix Using the Adjoint Method Inverse of a Matrix using Elementary Row Operations Applications of Inverse Matrices: Cryptography Some Properties of Inverses Input – Output Analysis Vectors Introduction Vector Spaces Spanning Set Basis and Dimensions

Week

Lecture Chapter 3

Notes

1

Lecture Chapter 4

Notes

2

Lecture Chapter 5

Notes

3

Lecture Chapter 6

Notes

4

Lecture Chapter 7

Notes

5

4 Dimension Linear Transformation Application of Linear Transformation Some Basic Theorems on Linear Transformations Linear Dependence of Vectors Solving Linear Equations

Lecture Chapter 8

Notes

6

Lecture Chapter 9

Notes

7&8

Lecture Chapter 10

Notes

9

Lecture Chapter 11

Notes

10

Lecture Chapter 12

Notes

11&12

Introduction Solving a system of simultaneous equations by the inverse method Economic Application Solving Linear Equations by Cramer’s Rule Proof of Cramer’s Theorem Economic Application Solving Linear Equations by the Gaussian Elimination Method Economic Application Linear Equations: Homogenous and Non – Homogenous Homogenous System of Equation Non – Homogeneous System of Equations Finding the General Solution Eigenvalues and Eigenvectors Introduction Characteristic Vectors Diagonalization Orthogonal Diagonalization Some Properties of Eigenvalues and Eigenvectors Symmetric and Skew-Symmetric Matrices Introduction Properties of Symmetric and Skew-Symmetric Matrices Quadratic Forms QF and positive definite matrices Linear Programming Introduction Constrained Maximization: Setting up a LP Model Extreme Point Theorem The Basis Theorem Constrained Minimization: Setting up a LP Model Linear programming: Simplex Method Simplex Algorithm for Maximization Problem Converting the Primal to a Dual Solving and Interpreting Results of the Dual

ADDITIONAL INFORMATION

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“Any candidate who has been absent from the University for a prolonged period during the teaching of a particular course for any reason other than illness or whose attendance at prescribed lectures, classes, ... tutorials, ... has been unsatisfactory or who has failed to submit essays or other exercises set by his/her teachers, may be debarred by the relevant Academic Board, on the recommendation of the relevant Faculty Board, from taking any University examinations. The procedures to be used shall be prescribed in Faculty Regulations.”

“97.

(i) (ii) (iii) (iv)

Cheating shall constitute a major offence under these regulations. Cheating is any attempt to benefit one’s self or another by deceit or fraud. Plagiarism is a form of cheating. Plagiarism is the unauthorized and/ or unacknowledged use of another person’s intellectual effort and creations howsoever recorded, including whether formally published or in manuscript or in typescript or other printed or electronically presented form and includes taking passages, ideas or structures from another work or author without proper and unequivocal attribution of such source(s), using the conventions for attributions or citing used in this University.

103.

(i)

If any candidate is suspected of cheating, or attempting to cheat, the circumstances shall be reported in writing to the Campus Registrar. The Campus Registrar shall refer the matter to the Chairman of the Campus Committee on Examinations. If the Chairman so decides, the Committee shall invite the candidate for an interview and shall conduct an investigation. If the candidate is found guilty of cheating or attempting to cheat, the Committee shall disqualify the candidate from the examination in the course concerned, and may also disqualify him/her from all examinations taken in that examination session; and may also disqualify him/her from all further examinations of the University, for any period of time, and may impose a fine not exceeding Bds$300.00 or J$5000.00 or TT$900.00 or US$150.00 (according to campus). If the candidate fails to attend and does not offer a satisfactory excuse prior to the hearing, the Committee may hear the case in the candidate’s absence.”

Course Activities Planned 1. Attendance at the Post National Budget Forum 2012, students are advised to check the Departmental Website (http://sta.uwi.edu/fss/economics/index.asp) after the presentation of the National Budget for 2012/13 in the Parliament. 2. Attendance at the Conference on the Economy (COTE2012) on October 11 – 12, 2012 students must register for attendance with the Economics Student Union (ESU) or the Department of Economics. How to study for this Course Students should keep up-to-date with lectures, and must attend tutorials. It is expected that the student reads Unit before the lecturer so that questions can be raised in the lecture on issues that

6 are still puzzling or where further clarification is required. Tutorial questions are to be attempted prior to the tutorial and ready for presentation during the session. Grading System GRADE A+ A AB+ B BC+ C CD+ D F

August 2013

GPA 4.3 4.0 3.7 3.3 3.0 2.7 2.3 2.0 1.7 1.3 1.0 0.0

MARKS% 86 and Over 70 -85 67 -69 63 -66 60- 62 57- 59 53 -56 50 -52 47 -49 43 -46 40 -42 0 - 39